Anca Iuliana Bonciocat, Nicolae Ciprian Bonciocat, Yann Bugeaud, Mihai Cipu, Maurice Mignotte: Apollonius circles and the number of irreducible factors of polynomials, 119-138

Abstract:

We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius circle associated to two points on the real axis with integer abscissae $a$ and $b$, with ratio of the distances to these points depending on the admissible divisors of $f(a)$ and $f(b)$. In particular, we obtain such upper bounds for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is an Eneström-Kakeya polynomial, or a Littlewood polynomial, or has a large leading coefficient. Similar results are also obtained for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.

Key Words: Apollonius circles, irreducible polynomial, prime number.

2020 Mathematics Subject Classification: Primary 11R09; Secondary 11C08.

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